The Supreme Chancellor is elected by and from the members of the Galactic Senate and serves as long as he can maintain the confidence of the house. He can be deposed by a vote of no confidence at any time.
The office was largely powerless in the last years of the Republic, similar in nature to a Speaker of the House. It was still a very prestigious position, however, and the Supreme Chancellor had a grand office and was given the title of "Excellency."
The office became much more powerful during the reign of Supreme Chancellor Palpatine, who was able to effectively force the Senate into granting him a series of "emergency" powers. Eventually, Palpatine became a dictator and proclaimed himself Emperor, thus abolishing the office of Supreme Chancellor.